Australian shares ended up 0.3 percent. They opened soft and moved up on expectations the central bank's policy meeting would leave its interest rate unchanged at 1.5 percent, which it did.

The Australian dollar jumped 0.5 percent to $0.762.

On Monday, U.S. share indexes posted losses of as much as 0.37 percent.

Tuesday's reversal in Asian shares "was a little bit surprising," said James Woods, global investment analyst at Rivkin in Sydney.

He attributed the changed course partly to views that Trump is now "making progress on his promises," citing movement toward repealing Obamacare.

"The other part is the technical aspect," Woods said. "Things are looking positive, we're seeing positive data reports, so investors are taking the dips as buying opportunities."

Republicans unveiled legislation to dismantle the Affordable Care Act on Monday, calling for an end to health insurance mandates and rolling back extra healthcare funding for the poor.

Congressional Democrats denounced the Republican plan, saying it would hurt Americans by requiring them to pay more for healthcare, to the benefit of insurers.

Trump's allegations over the weekend that he was wiretapped by his predecessor Barack Obama, without offering any evidence, also raised concerns about his ability to focus on his promised economic measures, including tax cuts and a boost to infrastructure spending.

The euro rose almost 0.1 percent to $1.0589, following a 0.4 percent slide on Monday after former French Prime Minister Alain Juppe ruled out standing in the country's presidential elections.

On Monday, European stocks posted losses, with Deutsche Bank's falling 7.9 percent, the biggest drag on the FTSEurofirst 300 index, after the German lender unveiled an 8 billion euro ($8.47 billion) cash call as part of a major reorganization.

But declines were limited by deals between French carmaker PSA Group and General Motors, and between Standard Life and Aberdeen Asset Management

The dollar index, which tracks the greenback against a basket of trade-weighted peers, was steady at 101.61, after Monday's 0.1 percent gain.

The dollar was also flat at 113.915, after Monday's 0.15 percent loss.

In commodities, U.S. oil pulled back 0.1 percent to $53.13 a barrel, following Monday's 0.2 percent drop, on ongoing concerns that U.S. production growth may undermine output cuts by the Organization of Petroleum Exporting Countries.